Effective mass and envelope approximation (II)

Generalising the procedure that we saw in section 4.2.1 to an electron and hole, we define a wavefunction in terms of a combined excitonic envelope function and Wannier functions

ψ(re, rh) =

X

Re,Rh

C(Re, Rh)aRe(re)aRh(rh), (4.8)

where the subscriptseandhdescribe electrons and holes respectively. This equation look slightly different to Eq. 4.2, because we have written the electron and holes as functions of their respective distances (re=relectron−Ri , rh =rhole−Ri). We can form an analogous equation to Eq. 4.7 and put in an explicit form forV(r)

~2 2m∗ e ∇2 Re + ~2 2m∗_h∇ 2 Rh− e2 4π|Re−Rh| C(Re, Rh) = EC(Re, Rh). (4.9)

Since the Coulomb term only in terms of the relative distance between the electron and hole, we can move into a centre of mass co-ordinate system

r=Re−Rh , R=

meRe+mhRh me+mh

− ~ 2M∇ 2 R C(R) = ERC(R) −~ 2 2µ∇ 2 r− e2 4πr C(r) =ErC(r). (4.12)

The first of these equations describes a free particle

ER= ~

2_K2

2M , (4.13)

where K =ke+kh. The second of the equations describes the hydrogen atom and so the total energy of the exciton is given by

E =Egap+~

2_K2 2M −

µe4

8h22. (4.14)

In direct band gap semiconductors the excitons form at K = 0 and so the energy equation, in this case, reduces to

E =Egap− µe4

8h22, (4.15)

which very nicely is just the result for hydrogen atom with an effective mass and charge screening shown by a dielectric constant added to the band gap energy.

In a quantum dot the confinement of the electrons can be smaller than the Bohr radius that we found for Mott-Wannier excitons. This is called the strong confine- ment regime. In this regime the electrons and hole can be thought of as independent particles, but masses of the electron and the hole are the effective masses for the con- duction (mc) and valence (mv) bands. They still interact with each other through the Coulomb interaction with a screening effect provided by the dielectric constant. A quantum dot is made of two materials with different band gaps and so you get

regions of confined states. Looking at Fig. 4.2, we see a cut through one of the spatial dimension and you would see the same thing if you cut though any of the others. Typically, the band gaps are bigger than the band offset,

Figure 4.2: Schematic diagram of the energy profile of a confined quantum dot.

Now we can return to the idea of a particle in a box. Let us think of a box of side length L and using the uncertainty principle, we may calculate the minimum confinement energy of the electrons to be

Ee,min = 9h2 8πmcL2

. (4.16)

From solving Schr¨odinger’s equation for a particle in a box, the discrete energy levels are given by Ee,n = (n2 x+n2y+n2z)h2 8mcL2 , Ee,min = 3h2 8mcL2 |nx,ny,nz=1, (4.17)

wherenx, ny, nz are integers, i.e., the principle quantum numbers in the three spacial dimensions andL as mentioned earlier is the size of the length box. The energy for holes is the same equation, but with the effective massmv. In addition to the particle energy, the electron sits above the band gap, which requires energy to achieve so

Ee,min =Egap+ 3h 2 8mcL2 , Eh,min = 3h 2 8mvL2 . (4.18)

The confinement of the system splits the heavy hole and light hole energies. This is due to the different curvatures of the two dispersions and their relation to the effective mass, _m1∗ =

d2_E

dk2. As can be readily seen by the equation, the lower the curvature, the higher the effective mass and so heavy holes have a greater effective mass than lighter holes. The energy of confined particles is proportional to _m1∗, where m∗ is the effective mass as shown in Eq. 4.2. This has the obvious consequence that heavy holes have a lower confinement energy than light holes. If we define the top of the valence band as having an energy of zero, then when an exciton is created, the electron in the conduction band will have a positive energy, the hole in the valence band will have a negative energy and the difference between them will be the energy of the absorbed photon. Since it takes less energy to confine a heavy hole than a light hole the energy gap between electron and hole will be smaller for a heavy hole than for a light hole, which is why heavy holes appear closer to the top of the valence band in Fig. 4.2.